If banks use the BTC blockchain in the future then of course it will effect the price. It'll send the price to the moon. If they use the BTC blockchain then it means they will have to temporarily convert currency into BTC whilst they transfer that money to wherever they're sending it. To transfer it they're buying BTC with the money they're sending so of course it will effect the price.
It'll be incredibly bullish if they start using the BTC blockchain, if they decide to use blockchain technology then the BTC blockchain is the only sensible way really, it's fast & incredibly secure, something banks will be have to ensure.
The banks would NOT have to temporarily convert currency into bitcoin. I could see them sending blocks of fed notes (1,000,000's 100,000's 10,000's) with marked satoshi's/colored coins. Of course they would still have to pay transaction fees with bitcoins. I believe it would still be bullish from Bitcoin being acknowledged. It just would be a way more subdued trip to the moon since they would not need a billion dollars of bitcoin to transfer billion dollars of fed notes just a few marked satoshi's.